Time-evolution of excitations in normal Fermi liquids

We inspect the initial and the long time evolution of excitations a Fermi liquids by analyzing the time behavior of the electron spectral function. Focusing on the short-time limit we study the electron-boson model for the homogenous electron gas and apply the first order (in boson propagator) cumulant expansion of the electron Green's function. In addition to a quadratic decay in time upon triggering the excitation, we identify non-analytic terms in the time expansion similar to those found in the Fermi edge singularity phenomenon. We also demonstrate that the exponential decay in time in the long-time limit is inconsistent with the GW approximation for the self-energy. The background for this is the Paley-Wiener theorem of complex analysis. To reconcile with the Fermi liquid behavior an inclusion of higher order diagrams (in the screened Coulomb interaction) is required.